Abstract
The concept of error matrix is presented in this paper, and the types of the fuzzy error matrix equation are presented too. The paper especially researches about the error matrix that consists of general set relations, and the solvability and solutions to it. And theorems about the necessary condition and the necessary and sufficient condition for the solvability of the error matrix equation XA′ = B are obtained in the paper. An example of solving this equation would be given in the last part of the paper.
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1 Introduction
Errors always happen in our life, work and study, sometimes with huge destructive effect, so it is necessary to study how to avoid and eliminate errors. And in order to avoid and eliminate errors, we have to study causes and laws of errors. And this paper discusses how to utilize the error matrix equation to eliminate errors.
2 The Concept of Error Matrix
Definition 1
Let
be an m × n error matrix of K elements.
Definition 2
Let
be a (t + 1) × 7 or m × 7 error matrix. Each element of this kind of error matrix is called a set.
Definition 3
The set relationship containing unknown sets is called set equations.
Definition 4
Let X, A′ and B be m × 7 error matrixes, so XA′ ⊇ (or other relational operators) B is named as set (matrix) equations.
3 Error Matrix Equation
Error matrix equation:
4 The Solution of Error Matrix Equation
The solution of Type II 1 XA = B
Definition 5
Let
and,
So the following equation is hold.
By the definition of equal matrices:if two matrices contain each other,so corresponding elements in both matrices contain each other. So \( \left( {w_{ij} ,z_{ij} } \right) \supseteq \left( {b_{ij} ,y_{ij} } \right) \),
So the following set equations are obtained:
About the operation symbol “∧”, if both sides of the equation are sets, then it means “intersection”, if both sides of the equation are numbers, then it means “minimum”.
As for (U 1ix ∧ U 2j) h 1 (S 1ix ∧ S 2j) h 2 (\( \overrightarrow {P}_{1ix} \wedge \overrightarrow {P}_{2j} \)) ∨ h 3 (T 1ix ∧ T 2j) h 4 (L 1ix ∧ L 2j) h 5 (x 1ix ∧ y 2j) h 6 (G U1ix ∧ G U2j), with “hi, i = 1, 2, …6”, means that elements have been computed could “compose” a complete matrix element (proposition). The mode of combination depends on different situations. One way is to constitute a new set of error elements or error logic proposition by parameter. And this way is called the multiplication of m × 7 error matrix.
Since what we required in solving practical problems is not XiA′ = B, but XiA′ ⊇ B, So we find a more general model of error matrix equation, that is to find out the solution of type II 1 equation XA′ ⊇ B.
Theorem 1
The sufficient and necessary condition for the solvability of error matrix equation XA′ ⊇ B is the solvability of X i A′ ⊇ B i , i = (1, 2, ……, m2).
Proof Suppose XA′ ⊇ B has solvability, it is can be known by the definitions of XA′ ⊇ B and XiA′ ⊇ Bi, i = (1, 2, …, m2) that they are the equivalent equations, so it is necessary for XiA′ ⊇ Bi, i = (1, 2, …, m2) has solvability; Otherwise, if the solvability of XiA′ ⊇ Bi, i = (1, 2, …, m2) exists, similarly it does for XA′ ⊇ B.
Thereout, we use the method of discussing the solvability of XiA′ = Bi, i = (0, 1, 2, …, m2) to discuss the solution of XA′ = B.
Then in XiA′ ⊇ Bi, we can get
Namely,
A series of set equations be obtained:
Theorem 2
The sufficient and necessary condition for the solvability of X i A′ ⊇ B i is,
Proof 1
If one of the conditions above is not satisfied, for example suppose S 2j (t) ⊇ S v2j(t) is not satisfied, so in the (S 1ix(t) ∧ S 2j (t)) = S v2j(t), no mater what value S 1ix(t) is, we can not get (S 1ix(t) ∧ S 2j (t)) = S v2j(t)。
Proof 2
Since we could only take union operation between the corresponding element of A and Xi in the XiA′ ⊇ Bi, that is
Then we discuss all the solutions of XiA′ ⊇ Bi and XA′ ⊇ B.
When X(x1, x2, xn) are obtained, we take intersection operation between X 与 Kg, rw, xq, so we can get X′(x′1, x′2, …, x′n) ∈ X′。
5 Examples of Error Matrix Equation
Suppose \( A^{\prime} = \left( {\begin{array}{*{20}c} {a_{11} } & {a_{12} } \\ {a_{21} } & {a_{22} } \\ \end{array} } \right) \), and
When n = 11, we get the following equations:
When n = 9, we get the following equations:
When n = 10, we get the following equations:
When n = 15, we get the following equations:
And suppose \( X = \left( {\begin{array}{*{20}c} {x_{1} } & {x_{2} } \\ \end{array} } \right) \), where
And suppose \( B^{\prime} = \left( {\begin{array}{*{20}c} {b_{11} } & {b_{12} } \\ {b_{21} } & {b_{22} } \\ \end{array} } \right) \), where
And when k = 7, we get the following equations:
And when k = 8, we get the following equations:
And when k = 2, we get the following equations:
And when k = 3, we get the following equations:
By the Theorem 2, when n = 11, we get the solution of XA′ ⊇ B, which is
Similarly when n = 8 we get the following equations:
6 Conclusion
We get the necessary and sufficient condition for the solvability of the error matrix equation XA′ = B, and they are also can be proved by the case studies.
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Acknowledgments
Thanks to the support by Guangzhou Vocational College of Science and Technology of China (No. 2015ZR01 and No. 2015JG08) and the provincial education system reform project of Department of Education of Guangdong Province, which gets the financial support form Department of Finance of Guangdong Province, with the number 16 in the third category of the second tier.
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Guo, Kz., Li, R., Li, Jx. (2016). Study of the Solvability of the Fuzzy Error Matrix Set Equation in Connotative Form of Type II 4. In: Cao, BY., Wang, PZ., Liu, ZL., Zhong, YB. (eds) International Conference on Oriental Thinking and Fuzzy Logic. Advances in Intelligent Systems and Computing, vol 443. Springer, Cham. https://doi.org/10.1007/978-3-319-30874-6_54
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DOI: https://doi.org/10.1007/978-3-319-30874-6_54
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